CTL,
The ticket proves that the person is entitled to the garment. The number also serves as an identifier both as to location in establishments that have many garments and to distinguish it from similar garments. In Israel one must always present the ticket when retrieving the garment from the cleaner – and the cleaners are invariably sabras.
In any case, many restaurants had a “no entrance without a tie” policy. There is a joke about someone who is walking through the desert when he finally sees someone. When he asks for water the guy says that he has none but is selling ties. The disappointed traveler continues until he sees a restaurant. When he cries “Water, water” the doorman says “No entrance without a tie”.